Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces

Abstract:

We present a measure-theoretic condition for a property to hold "almost everywhere" on an infinite-dimensional vector space, with particular emphasis on function spaces such as ${C^k}$ and ${L^p}$. Like the concept of "Lebesgue almost every" on finite-dimensional spaces, our notion of "prevalence" is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of "open and dense" or "generic" when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold "almost everywhere" in the sense of prevalence. For instance, we prove that almost every ${C^1}$ map on ${\mathbb {R}^n}$ has the property that all of its periodic orbits are hyperbolic.